Download 2.6 Linear Inequalities in Two Variables

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2.6
Linear Inequalities in Two
Variables
What you should learn
GOAL 1 Graph linear
inequalities in two variables.
GOAL 2 Use linear
inequalities to solve real-life
problems, such as finding the
number of minutes you can
call relatives using a calling
card in Example 4.
Why you should learn it
RE
GRAPHING LINEAR INEQUALITIES
A linear inequality in two variables is an inequality that can be written in one of
the following forms:
Ax + By < C,
Ax + By ≤ C,
Ax + By > C,
Ax + By ≥ C
An ordered pair (x, y) is a solution of a linear inequality if the inequality is true
when the values of x and y are substituted into the inequality. For instance, (º6, 2)
is a solution of y ≥ 3x º 9 because 2 ≥ 3(º6) º 9 is a true statement.
EXAMPLE 1
Checking Solutions of Inequalities
Check whether the given ordered pair is a solution of 2x + 3y ≥ 5.
a. (0, 1)
b. (4, º1)
c. (2, 1)
FE
To model real-life data,
such as blood pressures in
your arm and ankle
in Ex. 45.
AL LI
GOAL 1
SOLUTION
ORDERED PAIR
SUBSTITUTE
CONCLUSION
a. (0, 1)
2(0) + 3(1) = 3 ≥
/ 5
(0, 1) is not a solution.
b. (4, º1)
2(4) + 3(º1) = 5 ≥ 5
(4, º1) is a solution.
c. (2, 1)
2(2) + 3(1) = 7 ≥ 5
(2, 1) is a solution.
ACTIVITY
Developing
Concepts
Investigating the Graph of an Inequality
1
Copy the scatter plot.
2
Test each circled point to see whether it is a
solution of x + y ≥ 1. If it is a solution, color
it blue. If it is not a solution, color it red.
3
Graph the line x + y = 1. What relationship
do you see between the colored points and
the line?
4
Describe a general strategy for graphing an
inequality in two variables.
y
1
1
The graph of a linear inequality in two variables is the graph of all solutions of the
inequality. The boundary line of the inequality divides the coordinate plane into two
half-planes: a shaded region which contains the points that are solutions of the
inequality, and an unshaded region which contains the points that are not.
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Chapter 2 Linear Equations and Functions
x
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G R A P H I N G A L I N E A R I N E Q UA L I T Y
The graph of a linear inequality in two variables is a half-plane. To graph a linear
inequality, follow these steps:
STEP 1
Graph the boundary line of the inequality. Use a dashed line for < or >
and a solid line for ≤ or ≥.
STEP 2
To decide which side of the boundary line to shade, test a point not on
the boundary line to see whether it is a solution of the inequality. Then
shade the appropriate half-plane.
EXAMPLE 2
STUDENT HELP
Look Back
For help with inequalities
in one variable, see p. 42.
Graphing Linear Inequalities in One Variable
Graph (a) y < º2 and (b) x ≤ 1 in a coordinate plane.
SOLUTION
a. Graph the boundary line y = º2.
Use a dashed line because y < º2.
Test the point (0, 0). Because (0, 0)
is not a solution of the inequality,
shade the half-plane below the line.
b. Graph the boundary line x = 1.
Use a solid line because x ≤ 1.
Test the point (0, 0). Because (0, 0)
is a solution of the inequality, shade
the half-plane to the left of the line.
y
y
1
x≤1
(0,0)
1
1
x
(0,0)
2
x
y < 2
EXAMPLE 3
Graphing Linear Inequalities in Two Variables
Graph (a) y < 2x and (b) 2x º 5y ≥ 10.
STUDENT HELP
Study Tip
Because your test point
must not be on the
boundary line, you may
not always be able to use
(0, 0) as a convenient
test point. In such cases
test a different point,
such as (1, 1) or (1, 0).
SOLUTION
a. Graph the boundary line y = 2x.
Use a dashed line because y < 2x.
Test the point (1, 1). Because (1, 1)
is a solution of the inequality,
shade the half-plane below the line.
b. Graph the boundary line 2x º 5y = 10.
Use a solid line because 2x º 5y ≥ 10.
Test the point (0, 0). Because (0, 0) is
not a solution of the inequality, shade
the half-plane below the line.
y
y
1
(1, 1)
1
(0, 0)
1
1
y < 2x
x
x
2x 5y ≥ 10
2.6 Linear Inequalities in Two Variables
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GOAL 2
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Communication
USING LINEAR INEQUALITIES IN REAL LIFE
EXAMPLE 4
Writing and Using a Linear Inequality
You have relatives living in both the United States and Mexico. You are given a
prepaid phone card worth $50. Calls within the continental United States cost
$.16 per minute and calls to Mexico cost $.44 per minute.
a. Write a linear inequality in two variables to represent the number of minutes
you can use for calls within the United States and for calls to Mexico.
b. Graph the inequality and discuss three possible solutions in the context of the
real-life situation.
SOLUTION
PROBLEM
SOLVING
STRATEGY
United States
Mexico
United States
Mexico
Value of
•
+ rate • time ≤ card
time
rate
a. VERBAL
MODEL
LABELS
ALGEBRAIC
MODEL
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for extra examples.
United States rate = 0.16
(dollars per minute)
United States time = x
(minutes)
Mexico rate = 0.44
(dollars per minute)
Mexico time = y
(minutes)
Value of card = 50
(dollars)
0.16 x + 0.44 y ≤ 50
b. Graph the boundary line 0.16x + 0.44y = 50. Use a solid line because
0.16x + 0.44y ≤ 50.
Test the point (0, 0). Because (0, 0) is a solution of the inequality, shade the
half-plane below the line. Finally, because x and y cannot be negative, restrict
the graph to points in the first quadrant.
Possible solutions are points within the shaded region shown.
One solution is to spend 65 minutes
on calls within the United States and
90 minutes on calls to Mexico. The
total cost will be $50.
Calling Cards
Mexico time (min)
y
(65, 90)
(83, 83)
90
To split the time evenly, you could
spend 83 minutes on calls within the
United States and 83 minutes on calls
to Mexico. The total cost will be $49.80.
60
30
(150, 30)
0
0
100 200
300
x
United States time (min)
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Chapter 2 Linear Equations and Functions
You could instead spend 150 minutes
on calls within the United States and
only 30 minutes on calls to Mexico.
The total cost will be $37.20.
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GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
1. Compare the graph of a linear inequality with the graph of a linear equation.
2. Would you use a dashed line or a solid line for the graph of Ax + By < C?
for the graph of Ax + By ≤ C? Explain.
Tell whether the statement is true or false. Explain.
4
3. The point , 0 is a solution of 3x º y > 4.
3
4. The graph of y < 3x + 5 is the half-plane below the line y = 3x + 5.
Skill Check
✓
GRAPHING INEQUALITIES Graph the inequality in a coordinate plane.
5. x > 5
6. y < º4
9. y ≥ ºx + 7
13.
4
8. ºy ≥ 3
7. 3x ≤ 1
2
10. y > x º 1
3
11. 2x º 3y < 6
12. x + 5y ≤ º10
CALLING CARDS Look back at Example 4. Suppose you have relatives living
in China instead of Mexico. Calls to China cost $.75 per minute. Write and graph a
linear inequality showing the number of minutes you can use for calls within the
United States and for calls to China. Then discuss three possible solutions in the
context of the real-life situation.
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 942.
CHECKING SOLUTIONS Check whether the given ordered pairs are solutions
of the inequality.
14. x ≤ º5; (0, 2), (º5, 1)
15. 2y ≥ 7; (1, º6), (0, 4)
16. y < º9x + 7; (º2, 2), (3, º8)
17. 19x + y ≥ º0.5; (2, 3), (º1, 0)
INEQUALITIES IN ONE VARIABLE Graph the inequality in a coordinate plane.
18. x ≤ 6
19. ºx ≥ 20
10
20. 10x ≥ 3
21. º3y < 21
22. 8y > º4
23. y < 0.75
MATCHING GRAPHS Match the inequality with its graph.
24. 2x º y ≥ 4
A.
25. º2x º y < 4
B.
y
1
C.
y
1
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 14–17
Example 2: Exs. 18–23,
33–44
Example 3: Exs. 24–44
Example 4: Exs. 45–51
26. 2x + y ≤ 4
1
y
1
x
x
1
1
x
INEQUALITIES IN TWO VARIABLES Graph the inequality.
27. y ≤ 3x + 11
28. y > º4 º x
29. y < 0.75x º 5
30. 3x + 12y > 4
31. 9x º 9y > º36
3
2
32. x + y > 1
2
3
2.6 Linear Inequalities in Two Variables
111
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MATCHING GRAPHS Match the inequality with its graph.
33. x + y > 2
A.
34. x ≥ 2
35. y ≤ ºx + 2
B.
y
C.
y
1
1
1
x
y
1
1
1
x
GRAPHING INEQUALITIES Graph the inequality in a coordinate plane.
36. 9x º 2y ≤ º18
1
39. y ≥ x + 10
5
1
42. 6x ≥ ºy
3
45.
3
37. y < 3x º 4
38. 5x > º20
40. 4y ≤ º6
41. 2x + 3y > 4
43. 0.25x + 3y > 19
44. x + y < 0
HEALTH RISKS By comparing the blood pressure in your ankle with the
blood pressure in your arm, a physician can determine whether your arteries are
becoming clogged with plaque. If the blood pressure in your ankle is less than
90% of the blood pressure in your arm, you may be at risk for heart disease.
Write and graph an inequality that relates the unacceptable blood pressure in
your ankle to the blood pressure in your arm.
NUTRITION In Exercises 46 and 47, use the following information.
Teenagers should consume at least 1200 milligrams of calcium per day. Suppose you
get calcium from two different sources, skim milk and cheddar cheese. One cup of
skim milk supplies 296 milligrams of calcium, and one slice of cheddar cheese
supplies 338 milligrams of calcium. Source: Nutrition in Exercise and Sport
FOCUS ON
CAREERS
46. Write and graph an inequality that represents the amounts of skim milk and
cheddar cheese you need to consume to meet your daily requirement of calcium.
47. Determine how many cups of skim milk you should drink if you have eaten two
slices of cheddar cheese.
MOVIES In Exercises 48 and 49, use the following information.
You receive a gift certificate for $25 to your local movie theater. Matinees are $4.50
each and evening shows are $7.50 each.
48. Write and graph an inequality that represents the numbers of matinees and
evening shows you can attend.
49. Give three possible combinations of the numbers of matinees and evening shows
you can attend.
RE
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AL I
NUTRITIONISTS
INT
A nutritionist plans
nutrition programs and
promotes healthy eating
habits. Over one half of
all nutritionists work in
hospitals, nursing homes,
or physician’s offices.
NE
ER T
CAREER LINK
www.mcdougallittell.com
112
FOOTBALL In Exercises 50 and 51, use the following information.
In one of its first five games of a season, a football team scored a school record of
63 points. In all of the first five games, points came from touchdowns worth 7 points
and field goals worth 3 points.
50. Write and graph an inequality that represents the numbers of touchdowns and
field goals the team could have scored in any of the first five games.
51. Give five possible numbers of points scored, including the number of
touchdowns and the number of field goals, for the first five games.
Chapter 2 Linear Equations and Functions
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Test
Preparation
52. MULTI-STEP PROBLEM You want to open your own truck rental company. You
do some research and find that the majority of truck rental companies in your
area charge a flat fee of $29.99, plus $.29 for every mile driven. You want to
charge less so that you can advertise your lower rate and get more business.
a. Write and graph an equation for the cost of renting a truck from other truck
rental companies.
b. Shade the region of the coordinate plane where the amount you will charge
must fall.
c. To charge less than your competitors, will you offer a lower flat fee, a lower
rate per mile, or both? Explain your choice.
d. Write and graph an equation for the cost of renting a truck from your
company in the same coordinate plane used in part (a).
e. CRITICAL THINKING Why can’t you offer a lower rate per mile but a higher
flat fee and still always charge less?
★ Challenge
VISUAL THINKING In Exercises 53–55, use
y
the graph shown.
53. Write the inequality whose graph is shown.
54. Explain how you came up with the
inequality.
EXTRA CHALLENGE
1
x
1
55. What real-life situation could the first–
quadrant portion of the graph represent?
www.mcdougallittell.com
MIXED REVIEW
SCIENTIFIC NOTATION Write the number in scientific notation. (Skills Review, p. 913)
56. 10,000,000
57. 1,650,000,000
58. 203,000
59. 0.00067
60. 0.0000009
61. 0.0808
GRAPHING EQUATIONS Graph the equation. (Review 2.3 for 2.7)
5
62. y = x º 5
2
63. y = º5x º 1
1
64. y = ºx + 6
2
65. x º y = 4
66. 2x + y = 6
67. º4x + y = 4
WRITING EQUATIONS Write an equation of the line that passes through the
given points. (Review 2.4 for 2.7)
68. (2, 2), (5, 5)
69. (0, 7), (5, 1)
70. (º1, 6), (8, º2)
71. (3, 2), (3, º4)
72. (1, 9), (º10, º6)
73. (4, º8), (º7, º8)
74.
GARDENING The horizontal middle of the United States is at about 40°N
latitude. As a rule of thumb, plants will bloom earlier south of 40°N latitude
3
5
and later north of 40°N latitude. The function w = (l º 40) gives the number
of weeks w (earlier or later) that plants at latitude l°N will bloom compared with
those at 40°N. The equation is valid from 35°N to 45°N latitude. Identify the
domain and range of the function and then graph the function. (Review 2.1)
2.6 Linear Inequalities in Two Variables
113